Within triangle ABC, draw three segments parallel to the sides of the
triangle, each touching two sides. The three segments meet at one
point, and they are all the same length, x. Find the length of x given
the length of the sides of triangle ABC.

Two observers on points A and B of a national park see a beginning fire
on point C. Knowing that the angles CAB = 45 degrees, ABC = 105 degrees
and that the distance between points A and B is of 15 kilometers,
determine the distances between B and C, and between A and C.

As far as I know, a trapezoid is defined as a quadrilateral with exactly one set of parallel sides. However, a very highly regarded educator and textbook author recently argued that this definition is incorrect. His definition of a trapezoid is that it is a quadrilateral that has at least one pair of parallel sides. A square, therefore, would be considered a trapezoid. Is he correct or are thousands of books going to be published with the wrong definition?

How do you determine the radius of the circle that maximizes the area of an irregular
n-gon circumscribed on it? With the Pari computer algebra system, Doctor Vogler
approaches the question using numerical techniques such as Newton's Method and a
binary search, which suggests that no closed-form expression exists.

Given right triangles ABC and DCB with rt angles at B and C, triangle
ABC's hypotenuse 20 and triangle DCB's hypotenuse 30. The hypotenuses
intersect at point E, a distance of 10 from BC. Find the length of BC.